use a double angle formula to find the exact value of the fo

use a double angle formula to find the exact value of the following expression without using a calculator.

sin(2tan^-1(3/7))

Solution

sin(2 tan^-1(3/7))

tan^-1(3/7)= u

So we get sin (2u)= 2 sin u cos u

tan u= 3/7

opposite=3, adjacent=7

hypotenuse=sqrt(9+49)= sqrt 58

sin u= opposite/hypotenuse

sin u= 3/sqrt 58,

cos u= adjacent/hypotenuse

cos u= 7/sqrt 58

sin(2 u) = 2 sin u cos u = 2(3/sqrt58)(7 /sqrt58)= 42/58=21/29

Therefore sin(2 tan^-1(3/7)=21/29